introduction to trigonometry angles and radians (ma3a2): define an understand angles measured in...

Introduction to Trigonometry

Angles and Radians

(MA3A2): Define an understand angles measured in degrees and radians.

the rotating ray and the positive half of the x-axis

angle

a positive angle whose vertex is at the center of a circle

central angle

2 angles with a sum of 90 degrees

complementary angles

2 angles in standard position that have the same terminal side

coterminal angles

The beginning ray of an angular rotation; the positive half of the x-axis

for an angle in standard position

initial side

a clockwise measurement of an angle

negative degree measure

Moving around a circle toward the right; a negative rotation

Clockwise

a counterclockwise measurement of an angle

positive degree measure

Opposite to clockwise; moving around the circle toward the left; a

positive rotation

counterclockwise

An angle in standard position with a terminal side that coincides with one

of the four axes

quadrantal angle

One of the 4 regions into which the x- and y-axes divide the coordinate

plane

quadrant

A unit of angle measure

radian

The position of an angle with vertex at the origin, initial side on the positive x-axis, and terminal side in the plane

standard position

2 angles with a sum of 180 degrees

supplementary angles

The ending ray of an angular rotation; rotating ray

terminal side

1 full revolution

Obj: Find the quadrant in which the terminal side of an angle lies.

half a revolution

¼ of a revolution

Obj: Find the quadrant in which the terminal side of an angle lies.

triple revolution

Quad. IQuad. II

Quad. III Quad. IV

0◦/360°

90◦

180◦

270◦

III

III IV

0◦/360◦

90◦

180◦

270◦

In which quadrant does the terminal side of the angle lie?

EX: 53°

III

III IV

0◦/360◦

90◦

180◦

270◦

In which quadrant does the terminal side of each angle lie?

EX: 253°

III

III IV

0◦/360◦

90◦

180◦

270◦

EX: In which quadrant does the terminal side of each angle lie?

EX: -126°

III

III IV

0◦/360◦

90◦

180◦

270◦


EX: -373°

III

III IV

0◦/360◦

90◦

180◦

270◦


EX: 460°

III

III IV

0◦/360◦

90◦

180◦

270◦

ON YOUR OWN: In which quadrant does the terminal side of each angle lie?

1. 47° 2. 212° 3. -43°

4. -135° 5. 365°

III

III IV

0◦/360◦

90◦

180◦

270◦

ON YOUR OWN: In which quadrant does the terminal side of each angle lie?

1. 47° 2. 212° 3. -43°

4. -135° 5. 365°

I III IV

III I

ON YOUR OWN: Draw each angle.

7. -90°6. 45°

8. 225° 9. 405°

Θ (theta)

Lowercase Greek letters are used to denote angles

α (alpha)

β (beta)

γ (gamma)

EX: Θ = 60°.

Finding Coterminal Angles (angles that have the same terminal side):

EX: Θ = 790°.

Finding Coterminal Angles (angles that have the same terminal side):

1) Add and subtract from 360°, OR2) Add and subtract from 2∏.

EX: Θ = 440°.

EX: Θ = -855°.

1. Θ = 790°.

ON YOUR OWN: Find one positive and one negative coterminal angle.

Reference Angle: the acute angle formed by the terminal side and the closest x-axis.

EX: Find the reference angle for each angle. Θ = 115°.

**Reference angles are ALWAYS positive!!

**Subtract from 180° or 360°.


EX: Θ = 225°.




EX: Θ = 330°



EX: Θ = -150°.

Finding Reference Angles fpr Negative Angles:

1) Add 360° to find the coterminal angle.2) Subtract from closest x-axis (180° or 360°) .

EX: Θ = 60°.

**If Θ lies in the first quadrant, then the angle is it’s own reference angle.

11. Θ = 210 °10. Θ = 405°

12. Θ = -300 ° 13. Θ = -225 °

ON YOUR OWN: Find the reference angle for the following angles of rotation.

Converting Degrees to Radians

*Multiply by ∏ radians 180 degrees

EX: 60°

EX: 225°

EX: 300°

EX: -315°

ON YOUR OWN: Convert to radian measure. Give answer in terms of ∏.

14. 15°

15. 30°

16. 100°

17. -220°

18. -85°

ON YOUR OWN: Convert to radian measure. Give answer in terms of ∏.

14. 15°

15. 30°

16. 100°

17. -220°

18. -85°

12

6

9

5

9

11

36

17

Converting Radians to Degrees

*Multiply by 180 degrees ∏ radians

EX:

EX:

EX:

EX:

4

33

4

2

55

4

ON YOUR OWN: Convert to Degrees

6

719.

20.

21.

22.

23.

12

4

3

4

5

2

3

-210°

-2160°

135°

225°

270°

ON YOUR OWN: Convert to Degrees

6

719.

20.

21.

22.

23.

12

4

3

4

5

2

3

Finding Coterminal Angles

*add or subtract from 360°*_____degrees ± 360

EX: 390° EX: 140°

EX: -100°

Finding Complements and Supplements

*To find the complement: subtract from 90°*To find the supplement: subtract from 180°

EX: 35° EX: 120°

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